Ebook: Dimensions of Ring Theory
- Tags: Commutative Rings and Algebras, Associative Rings and Algebras, Non-associative Rings and Algebras
- Series: Mathematics and Its Applications 36
- Year: 1987
- Publisher: Springer Netherlands
- Edition: 1
- Language: English
- pdf
Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Gad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of s9phistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-2
Finiteness Conditions for Lattices....Pages 3-36
Finiteness Conditions for Modules....Pages 37-120
Krull Dimension and Gabriel Dimension of an Ordered Set....Pages 121-142
Krull Dimension and Gabriel Dimension of Rings and Modules....Pages 143-170
Rings with Krull Dimension....Pages 171-190
Krull Dimension of Noetherian Rings. The Principal Ideal Theorem....Pages 191-215
Relative Krull and Gabriel Dimensions....Pages 217-247
Homological Dimensions....Pages 249-292
Rings of Finite Global Dimension....Pages 293-312
The Gelfand-Kirillov Dimension....Pages 313-342
Back Matter....Pages 343-360
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-2
Finiteness Conditions for Lattices....Pages 3-36
Finiteness Conditions for Modules....Pages 37-120
Krull Dimension and Gabriel Dimension of an Ordered Set....Pages 121-142
Krull Dimension and Gabriel Dimension of Rings and Modules....Pages 143-170
Rings with Krull Dimension....Pages 171-190
Krull Dimension of Noetherian Rings. The Principal Ideal Theorem....Pages 191-215
Relative Krull and Gabriel Dimensions....Pages 217-247
Homological Dimensions....Pages 249-292
Rings of Finite Global Dimension....Pages 293-312
The Gelfand-Kirillov Dimension....Pages 313-342
Back Matter....Pages 343-360
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